Scattering Theory of Gilbert Damping

Abstract

The magnetization dynamics of a single domain ferromagnet in contact with a thermal bath is studied by scattering theory. We recover the Landau-Liftshitz-Gilbert equation and express the effective fields and Gilbert damping tensor in terms of the scattering matrix. Dissipation of magnetic energy equals energy current pumped out of the system by the time-dependent magnetization, with separable spin-relaxation induced bulk and spin-pumping generated interface contributions. In linear response, our scattering theory for the Gilbert damping tensor is equivalent with the Kubo formalism. DOI: 10.1103/PhysRevLett.101.037207 PACS numbers: 75.40.Gb, 72.25.Mk, 76.60.Es Magnetization relaxation is a collective many-body phenomenon that remains intriguing despite decades of theoretical and experimental investigations. It is important in topics of current interest since it determines the magnetization dynamics and noise in magnetic memory devices and state-of-the-art magnetoelectronic experiments on current-induced magnetization dynamics [1]. Magnetization relaxation is often described in terms of a damping torque in the phenomenological Landau-Lifshitz-Gilbert (LLG) equationwhere M is the magnetization vector, g B =@ is the gyromagnetic ratio in terms of the g factor and the Bohr magneton B , and M s jMj is the saturation magnetization. Usually, the Gilbert dampingG M is assumed to be a scalar and isotropic parameter, but in general it is a symmetric 3 3 tensor. The LLG equation has been derived microscopically [2] and successfully describes the measured response of ferromagnetic bulk materials and thin films in terms of a few material-specific parameters that are accessible to ferromagnetic-resonance (FMR) experiments [3]. We focus in the following on small ferromagnets in which the spatial degrees of freedom are frozen out (macrospin model). Gilbert damping predicts a strictly linear dependence of FMR linewidths on frequency. This distinguishes it from inhomogenous broadening associated with dephasing of the global precession, which typically induces a weaker frequency dependence as well as a zerofrequency contribution.The effective magnetic field H eff ÿ@F=@M is the derivative of the free energy F of the magnetic system in an external magnetic field H ext , including the classical magnetic dipolar field H d . When the ferromagnet is part of an open system as in Fig. 1, ÿ@F=@M can be expressed in terms of a scattering S matrix, quite analogous to the interlayer exchange coupling between ferromagnetic layers [4]. The scattering matrix is defined in the space of the transport channels that connect a scattering region (the sample) to thermodynamic (left and right) reservoirs by electric contacts that are modeled by ideal leads. Scattering matrices also contain information to describe giant magnetoresistance, spin pumping and spin battery, and current-induced magnetization dynamics in layered normal-metal N j ferromagnet (F) systems [4 -6].In the following we demonstrate that scattering theory can be also used to compute ...

Ab InitioCalculation of the Gilbert Damping Parameter via the Linear Response Formalism

“…As pointed out by Brataas et al [7], their approach is completely equivalent to a formulation in terms of the linear response or Kubo formalism. The latter route is taken in this communication that presents a Kubo-Greenwoodlike expression for the Gilbert damping parameter.…”

“…However, in practice, like many other models it depends on a relaxation time parameter τ that describes the rate of transfer due to the various types of possible scattering mechanisms. This weak point could be removed recently by Brataas et al [7] who described the Gilbert damping by means of scattering theory. This development supplied the formal basis for the first parameter-free investigations on disordered alloys for which the dominant scattering mechanism is potential scattering caused by chemical disorder [8].…”

Nonlocal torque operators inab initiotheory of the Gilbert damping in random ferromagnetic alloys

“…[9][10][11][12]14,20 The dissipation of magnetic energy accompanying the slow magnetization dynamics, evaluated within a scattering theory or the Kubo linear response formalism, leads also to explicit expressions for the Gilbert damping tensor. 13,[17][18][19] Most of these formulations yield relations equivalent to the so-called torquecorrelation formula…”

“…A large number of theoretical approaches to the Gilbert damping has been worked out during the last two decades; here we mention only schemes within the oneelectron theory of itinerant magnets, [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] where the most important effects of electron-electron interaction are captured by means of a local spin-dependent exchangecorrelation (XC) potential. These techniques can be naturally combined with existing first-principles techniques based on the density-functional theory, which leads to parameter-free calculations of the Gilbert damping tensor of pure ferromagnetic metals, their ordered and disordered alloys, diluted magnetic semiconductors, etc.…”

Ultra-low magnetic damping of a metallic ferromagnet

“…More recently, Gilmore and Stiles 2 as well as Thonig et al 12 demonstrated a generalized torque correlation model that includes both intraband (conductivity-like) and interband (resistivity-like) transitions. The use of scattering theory to describe damping was later applied by Brataas et al 13 and Liu et al 14 to describe damping in transition metals. A numerical realization of a linear response damping model was implemented by Mankovsky 3 for Ni-Co, Ni-Fe, Fe-V and Co-Fe alloys.…”